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THURSDAY, SEPTEMBER 27, 1S94. 



THE WORKS OF HENRY J. S. SMITH. 

 ':t' Collected Mathematical Papers of Henry John 

 Stephen Smith. 2 vols. Edited by Dr. J. \V. L. 

 Glaisher. (Oxford : Clarendon Press, 1894.) 



rHE long looked for collected papers of Prof. H. J. S. 

 Smith, late Savilian Professor of Geometry in the 

 niversity of Oxford, have now appeared in two hand- 

 • me quarto volumes issued by the Clarendon Press at 

 '.ford. This fact is, as far as England is concerned, 

 he mathematical event of the year, and is of the utmost 

 inportance to mathematicians in general, and to the 

 ;ng race of investigators in pure mathematics in par- 

 iilar. The work has a portrait on the frontispiece, 

 nd is introduced by a biographical sketch by Dr. Charles 

 I. Pearson, and recollections by Prof. Jowett, Lord 

 owen, Mr. J. L. Strachan-Davidson, and Mr. Alfred 

 ibinson, also by an introduction by Dr. J. \V L. 

 lisher. A perusal of the sketch is calculated to greatly 

 iipress the reader with the all-round scholarship and 

 •ellectual eminence of its subject. To have gained, 

 niongst other honours, the Ireland University Scholar- 

 hip, and subsequently to have become one of the most 

 rofound and rigorously exact mathematicians the world 

 IS ever known, implies the possession of powers of mind 

 Lit must fill any chronicler or student of past events 

 ith amazement. There are men who will succeed in 

 ny line of life or branch of study by sheer mental 

 ength ; they have the faculty of becoming fascinated 

 any pursuit in which inclination, force of circum- 

 mces, or accident leads them to engage ; with them 

 Jyis intense concentration leading, through a flood of 

 ew ideas, to such an admiration for and interest in the 

 bject, of whatever kind, as can only be experienced by 

 ■ •.her in some special branch for which his mind is 

 irticularly and peculiarly adapted. 

 That Prof. Smith was such a man, was the general 

 icf of his contemporaries. Prof. Conington said to 

 ' biographer, " I do not know what Henry Smith may be 

 ■ the subjects of which he professes to know something ; 

 It I never go to him about a matter of scholarship, in a 

 1: where he professes to know nothing, without learning 

 ue from him than I can get from anyone else." 

 .\t one time it appeared to be probable that he would 

 ■\ote himself to chemical science, but, looking back, 

 re seems to be little doubt that pure mathematics was 

 le branch of knowledge towards which he felt himself 

 lost attracted, and which was in reality best adapted 

 1 call forth his grand powers for close and accurate 

 hinking, and to give scope to his brilliant imagination. 



The recollections are of great interest. They show 

 learly the extent to which he was admired and loved by 

 hose who were privileged to know him best. Dr. 

 •iaisher's introduction is chiefly, though not wholly, of 

 nathematical interest, and will be further alluded to. 

 The works set forth were published between the years 

 S51 and 1SS3. They may be considered as arranging 

 hemselves under four heads: (i) Theory of numbers ; 

 -) elliptic functions ; (3) geometry ; and (4) addresses. 

 >pace merely permits me to note some of the original 



contributions to science which stand forth pre-eminently, 

 and helped to build up a great reputation. 



During the years 1859 1S65 was produced the" Report 

 on the Theory of Numbers," compiled for the British 

 Association for the Advancement of Science. Prof. 

 Smith said of Clifford that he was " above all and before 

 all a geometer " ; so of him it may be said that he was 

 above all and before all an arithmetician, and that the 

 Report could not have come under a stronger hand. It 

 contains an account, confessedly not exhaustive, of the 

 state of knowledge at the date of writing. There is inter- 

 polated in the history of the science, as it was originated 

 by Gauss and Legendre, and developed by Cauchy, Jacobi, 

 Lejeune-Dirichlet, Eisenstein, Poinsot, Kummer, Kron- 

 ecker, and Hermite, a considerable amount of masterly 

 criticism as well as original work. He considers the higher 

 arithmetic to be comprised of two principal branches, 

 the theory of congruencies and the theory of homogeneous 

 forms. It will be observed that he does not include the 

 combinational or partitional analysis. He doubtless did 

 not regard this important subject as a branch of 

 arithmetic proper, but rather as occupying the ground 

 intermediate to arithmetic and algebra. It is, in point of 

 fact, far less abstruse and less dependent upon methods 

 which are regarded as purely arithmetical. In the future, 

 however, it is probable that it will be recognised that the 

 combinational analysis is able to throw quite unexpected 

 light on the theory of congruencies, and is worthy of 

 being considered as .in important instrument of research 

 in arithmetic proper. As an example it may be stated 

 that the enumeration of certain permutations on a circle 

 yields the number 



n 



n 

 where x and 71 are any positive integers, <^a division of ;;, 

 and (^ {d) the totient of n ; and hence 

 « 

 2 (j) (d) x"^ = o mod u 



a congruence which includes several of the elementary 

 results of the theory of numbers. The author's inten- 

 tion was to present the theory of homogeneous forms in 

 the following order : — (i) Binary quadratic forms ; (2) 

 binary cubic forms ; (3) other binary forms ; 4) ternary 

 forms ; (5) other quadratic forms ; (6) forms of order 11 

 decomposable into n linear factors. It is much to be 

 regretted that only the first of these was given in the 

 report. A consideration of the remaining divisions 

 seems to have convinced him that much remained to be 

 done, and he appears to have deferred these matters for 

 future investigation by himself. The solution of the so- 

 called " Pellian Equation " is of primary importance in 

 the theory of quadratic forms of positive and not square 

 determinant, and we find in the foot-note of p. 193, vol. i. 

 " There does not seem to be any ground for atfibuting 

 either the problem or its solution to Pell." This is par- 

 ticularly interesting to those who were privileged to listen 

 to Prof. Mittag Leftler's paper on automorphic functions, 

 which was read before Section A of the British Asso- 

 ciation at Oxiord. The I'rofessor inveighed against the 

 too common practice of associating mathematicians' 

 names with theories and theorems on the ground that 

 mistakes are of frequent occurrence and necessarily so ; 



NO. I30O: VOL. 50] 



